[Math] Find an invertible matrix U such that UA = R is in reduced row-echelon form , and express U as a product of elementary matrices.

matrices

Find an invertible matrix $U$ such that $UA = R$ is in reduced row-echelon form , and express $U$ as a
product of elementary matrices.

A

I have found R(RREF of $A$), but cannot do $U = R * A$ (cannot multiply 4×3 on 4×3)

Best Answer

I use

  • $E_i(c)$ to denote the elementary operation of “multiplying the $i$-th row by $c$” (for $c\ne0$);

  • $E_{ij}(d)$ to denote the elementary operation of “adding to the $i$-th row the $j$-th row multiplied by $d$” (for $i\ne j$);

  • $E_{ij}$ to denote the elementary operation of “switching the $i$-th and $j$-th rows (for $i\ne j$).

This is convenient because doing the elementary operation on rows is equivalent to multiplying by a matrix, the one obtained by applying the elementary operation to the identity matrix. Such matrices are denoted with the same symbol and are known as elementary matrices.

The operations you have to perform on your matrix to get the RREF are, in order, $$ E_1(1/2)\quad E_{21}(-3)\quad E_{31}(-1)\quad E_{2}(-2/5)\quad E_{32}(5/2)\quad E_{12}(-1/2) $$ and the RREF form is $$ R=\begin{bmatrix} 1 & 0 & 7/5 & 1/5 \\ 0 & 1 & -7/5 & -2/5 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$ Thus we know that $$ R= E_{12}(-1/2) E_{32}(5/2) E_{2}(-2/5) E_{31}(-1) E_{21}(-3) E_1(1/2) A $$

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